Realizing symmetry-guaranteed pairs of bound states in the continuum in metasurfaces

Bound states in the continuum (BICs) have received significant attention for their ability to enhance light-matter interactions across a wide range of systems, including lasers, sensors, and frequency mixers. However, many applications require degenerate or nearly degenerate high-quality factor (Q) modes, such as spontaneous parametric down conversion, non-linear four-wave mixing, and intra-cavity difference frequency mixing for terahertz generation. Previously, degenerate pairs of bound states in the continuum (BICs) have been created by fine-tuning the structure to engineer the degeneracy, yielding BICs that respond unpredictably to structure imperfections and material variations. Instead, using a group theoretic approach, we present a design paradigm based on six-fold rotational symmetry (C6) for creating degenerate pairs of symmetry-protected BICs, whose frequency splitting and Q-factors can be independently and predictably controlled, yielding a complete design phase space. Using a combination of resonator and lattice deformations in silicon metasurfaces, we experimentally demonstrate the ability to tune mode spacing from 2 nm to 110 nm while simultaneously controlling Q-factor.

The authors of manuscript "Realizing symmetry-guaranteed pairs of bound states in the continuum in metasurfaces" suggest a design of metasurface supporting bound states in the continuum (BIC) protected by symmetry. Based on the group theoretical analysis they consider a high-symmetry structure supporting BIC. Reducing of different type of symmetry allows one to control either quality (Q) factor or mode splitting. I like the underlying idea very much and in general it suitable to be a publication in NCOMM. In the present form it is really far of it, though.
1. Authors do not reveal they distinguish point symmetry of scatterer and lattice symmetry. For example, the transformation of the disk shape into triangle they consider as C2 breaking. At the same time the lattice deformation they treat as C3 breaking. Although they really are, these two modifications of the metasurface have very different nature. I easily imagine change of the scatterer shape that reduces any of the considered symmetry and the same reduction related to the lattice deformation as well. I expected that the former leads to the control of Q factor, and the latter is responsible to the mode splitting. I believe it is the most important results of the manuscript and these two effects authors have to elaborate in great details and illustrations which mechanism results in which effect.
2. Starting from Section "Independent control of splitting and Q-factor" theoretical and experimental results are described in confusion way. A small part of text is to experiment and the very next part is theory without any warning and right at once it goes back to the experiment and so on. It misleads a reader and has to be rewritten in a more structuralized way.
3. Oblique incidence is known to cause reduction of the symmetry. Is a one degree convergence is negligible. The authors have to carry out proper simulations sustaining this assumption.
4. The manuscript contains vast of the group-symmetry terms, which are in excessive way and sometimes misleading. I recommend the authors to give their manuscript to an expert in the group theory analysis applied to condensed matters to improve the terminology. 5. Several times the authors claim on benefits over the problem of fabrication imperfections, however they refer to no studies. For example, recent paper in Nanophotonics 10, 4313 (2021) examines the robustness of the BIC of different type against the typical fabrication imperfectness, that is positional disorder. I recommend the authors to consider that paper and references therein.
Overall, I like the idea but the present form of the manuscript makes me unhappy. My recommendation is a major revision mainly to improve the presentation of the results and to make it clearer.
In the manuscript, Authors discuss the origin of bound states in the continuum, high-Q resonances of periodic photonic structures, in terms of group theory. In particular, they present an approach to create two degenerate symmetry-protected bound states in the continuum and control their quality factor. It is a beautiful example of rigorous theoretical analysis which results can be implemented in practical applications.
Please, find comments and questions below.
Line 145 seems to have a misprint.
In Fig.3(a) diagram shows different regimes of symmetry breaking. How was the quality factor obtained? Highly-likely, from simulation, but why did the Authors take average values of quality factor?
Could the Authors comment on how the orientation of the axis along which the lattice is squeezed, is chosen? Is it restricted by basis vectors of the primitive unit cell? Will there be any quantitative differences in Qs and frequencies splitting compared to the discussed case? Since no light can enter or leave a BIC, designing a system to operate at a BIC is generally undesirable for most applications. However, the existence of a BIC in a system's parameter space guarantees a neighborhood in design space (around the BIC) where a quasi-BIC is formed and the state's Qfactor can be made arbitrarily large. While there are a few different methods for creating BICs in metasurfaces 7−12 :::: 7−14 , protecting a state from radiating due to its incompatible symmetry with the radiative channels in free space has proven to be a robust experimental pathway for creating :::::::: generating BICs.

Comment #2
RC: The authors state in page 3 (lines 69-72) that "as our methodology is based solely on symmetry and does not rely upon fine-tuning the states of a structure, it is robust against material variations and fabrication imperfections; any changes in the filling fraction (fabrication imperfection) or dielectric constant (material variation) of the metasurface affects the two states equally, and does not result in uncontrolled changes to the states Q-factors or frequency splitting." However, first of all material fluctuations break the symmetry C 3v and C 3 that removes degeneracy of BICs classified according to irreducible presentations of group symmetry. It would nice if the authors argue their statements more clearly. Anyway I believe the structural imperfections are to be small enough in order to guarantee pairs of degenerate quasi-BICs.
AR: Thank you for identifying this compact wording resulting in a loss of clarity. Both fabrication and material variations can result in symmetry preserving imperfections or symmetry breaking imperfections. We intended to highlight the fact that there are many types of common material and fabrication imperfections that do not break C 3v or C 3 symmetries, including resonator thickness, resonator diameter, lattice periodicity, and isotropic refractive index. In comparison, when designing accidental degeneracies through fine-tuning the degeneracy will be susceptible to these imperfections in general. We have expanded the discussion in page 3 (lines 69-72) to more clearly articulate the comparison between symmetry-guaranteed and accidental degeneracies.

Comment #3
RC: Why the simple rotational groups C2 and C3 are considered by the authors as nontrivial groups. Any group Cn is the Abel group with one-dimensional irreducible representations. May be the authors imply the group C3v which has one two-dimensional irreducible representation E and therefore modes can be two-fold degenerate. Respectively, already they can be symmetry-guaranteed pairs of the BICs.
AR: We appreciate the reviewer pointing out our lack of clarity on this issue. Yes, any C n group, by itself, will only possess one-dimensional irreducible representations. However, our structures use materials that are time-reversal symmetric (indeed, most materials used in nanophotonic systems have this property). Thus, the the full set symmetries of our system are described by the nonunitary groupḠ = G + T G, that combines both the unitary space group operations with the antiunitary operations formed by the composition of a space group operation with T , the time-reversal operator. The inclusion of time-reversal symmetry can guarantee additional degeneracies in a system's spectrum that are not present when only considering the unitary space group operators, see Chapter 13 of Ref. [1].
The net effect of time-reversal symmetry for our photonic (bosonic) crystal slab systems is to force the same degeneracies that are found in slab systems with reflection symmetries. In other words, both a photonic crystal slab with C 3v symmetry, and a slab with T and C 3 symmetries, will possess a symmetry guaranteed degeneracy between time-reversal conjugate representations that are otherwise one-dimensional representations.
To clarify these concepts, we have substantially amended the discussion of symmetries in our manuscript.

Comment #4
RC: Could the authors present more solid arguments in favor of that the degenerated BICs wide limits the classes of non-linear phenomena?
AR: We thank the reviewer for identifying this weak argument. When designing nearly degenerate high-Q states there are four properties to control: center frequency, mode spacing, Q-factors, and polarization states. When using accidentally degenerate symmetry-protected BICs only the Q-factors are guaranteed to be protected by symmetry. This creates a design space that is very susceptible to symmetry preserving and symmetry breaking imperfections resulting in an experimental platform that while workable in theory, can become extremely challenging in practice. With symmetry-guaranteed pairs of symmetry-protected BICs, not only are Q-factors protected, but also mode spacing and relative polarization states (the modes are guaranteed to have orthogonal polarization states) leaving only the center frequency unprotected. These additional symmetry protections greatly simplify the design space and feasibility of realizing nearly degenerate high-Q states, especially for applications such as spontaneous parametric down-conversion (SPDC) where ideally all four properties would be protected [2]. We have expanded the discussion in the introduction to more clearly articulate this argument. The modified section is reproduced below for the reviewer's convenience.

Reviewer #2
RC: The authors of manuscript "Realizing symmetry-guaranteed pairs of bound states in the continuum in metasurfaces" suggest a design of metasurface supporting bound states in the continuum (BIC) protected by symmetry. Based on the group theoretical analysis they consider a high-symmetry structure supporting BIC. Reducing of different type of symmetry allows one to control either quality (Q) factor or mode splitting. I like the underlying idea very much and in general it suitable to be a publication in NCOMM. In the present form it is really far of it, though.
AR: We want to thank the reviewer for their time and constructive feedback. The feedback enabled us to significantly strengthen the manuscript.

Comment #1
RC: Authors do not reveal they distinguish point symmetry of scatterer and lattice symmetry. For example, the transformation of the disk shape into triangle they consider as C2 breaking. At the same time the lattice deformation they treat as C3 breaking. Although they really are, these two modifications of the metasurface have very different nature. I easily imagine change of the scatterer shape that reduces any of the considered symmetry and the same reduction related to the lattice deformation as well. I expected that the former leads to the control of Q factor, and the latter is responsible to the mode splitting. I believe it is the most important results of the manuscript and these two effects authors have to elaborate in great details and illustrations which mechanism results in which effect.
AR: We thank the reviewer for identifying the lack of clarity in the manuscript to describe the physical mechanism used in the manuscript. For this work we consider the symmetry of the scatterer and lattice together, not separately. If we understand the reviewer correctly, they are making two points in this comment. First, that it is possible to use either the scatterer's shape or the lattice deformation to completely reduce the system's symmetry, or to otherwise control the system's symmetry. We agree with this comment, we simply chose to use the scatterer's shape and lattice deformation together for simplicity of experimental realization.
Second, the reviewer is hypothesizing that, rather than C 2 -breaking and C 3 -breaking controlling the Q-factors and mode splitting, respectively, that the Q-factor is instead completely controlled by the scatterer's geometry (regardless of what symmetries it possesses), and the mode splitting is completely controlled by the lattice deformation. On this point, we disagree.
To clearly demonstrate that it is the two different types of symmetry-breaking that provide independent control over the modal Q-factors and frequency splitting, we have updated the manuscript and supplementary information to include simulated and/or experimental results for a total of eight cases demonstrating C 2 -breaking and C 3 -breaking controlling the Q-factors and mode splitting using scatterer and/or lattice deformations. In particular, these simulations confirm that C 2 -breaking always controls the modes' Q-factors, regardless of whether C 2 is broken by the scatterer's shape or a deformation of the lattice, and C 3 -breaking always controls the frequency splitting between the two resonances, again, regardless of whether changes in the scatterer or lattice is responsible for this change in the system's symmetry.
To improve the clarity of the paper we have added a discussion to the manuscript to more clearly state the role of C 2 and C 3 symmetry breaking operations. Furthermore, we have modified the text to more clearly state why both resonator and lattice deformations were used to control Q-factor and mode splitting respectively. Finally, we have modified the manuscript to more clearly reference the extended group theory discussion, experimental demonstration using only resonator deformations, and additional numerical simulations in the supplementary information. The added sections, modified sections, and relating figures have been reproduced here for the reviewer's convenience.

Resonator deformations to control mode spacing
Instead of using lattice deformations to break C 3 and control the mode spacing, it is possible to use resonator deformations. To illustrate this we performed full-wave eigenvalue simulations when deforming cylindrical resonators to square prisms as depicted in Fig. S7a. Through this deformation, C 6 and C 3 symmetries are broken, which yields a lattice that belongs to the cmm space group, and results in the symmetry-protected degenerate BICs becoming non-degenerate BICs. Correspondingly, as the symmetry breaking parameter is increased the mode splitting increases as shown in Fig. S7b, but since both modes are BICs the Q-factors remain ≈ ∞ for all symmetry breaking parameters as confirmed in Fig. S7c. C 6v , C 3v , C 2v , C 2 , C s Figure R1: Circle to square deformation to control splitting (a) Schematic illustrating deforming the cylindrical resonators in a triangular lattice to square prism resonators. This breaks C 6 and C 3 symmetries reducing the the symmetry from a lattice belonging to the p6mm space group to one belonging to the cmm space group causing the symmetry-protected degenerate BICs to become non-degenerate BICs. This can be observed from the mode wavelengths (b) and Q-factors (c) as a function of symmetry breaking. As the the cylindrical resonators are deformed into square prisms the degeneracy is lifted. But since the modes are non-degenerate BICs, the modes remain BICs with Q-factors ≈ ∞ for all symmetry breaking parameters.

Lattice deformations to control mode spacing and Q-factors
To show a single lattice deformation capable of controlling mode spacing and Q-factors we performed full-wave eigenvalue simulations when the symmetry breaking operation was shifting alternating rows of resonators. A schematic of this symmetry breaking operation is presented in Fig. S8a. Shifting alternating rows of resonators causes all symmetries to be broken resulting in a lattice belonging to the p1 space group. This symmetry reduction, results in the symmetry-protected degenerate BICs in becoming non-degenerate quasi-BICs. This behavior is confirmed by the full-wave eigenvalue simulations. As the resonator deformation is increased from 0 nm the degeneracy is lifted (Fig. S8b) 6 and the Q-factors (Fig. S8c) Figure R3: Shifting alternating rows of resonators to control splitting and Q-factors (a) Schematic illustrating shifting alternating rows of cylindrical resonators. This deformation breaks all symmetries reducing the the symmetry from a lattice belonging to the p6mm space group to one belonging to the p1 space group causing the symmetry-protected degenerate BICs to become non-degenerate quasi-BICs. This effect can be observed in the wavelengths (b) and Q-factors (c). As expected, when the displacement is nonzero the degeneracy is lifted and the Q-factors become finite.

Comment #2
RC: Starting from Section "Independent control of splitting and Q-factor" theoretical and experimental results are described in confusion way. A small part of text is to experiment and the very next part is theory without any warning and right at once it goes back to the experiment and so on. It misleads a reader and has to be rewritten in a more structuralized way AR: We thank the reviewer for identifying how the structure unnecessarily reduces clarity. To improve clarity we have updated the manuscript and changed the structure of the section "Independent control of splitting and Q-factor" to move from design to simulations and ending with experimental results. Because of the long length of the section we have not reproduced the updated section below.

Comment #3
RC: Oblique incidence is known to cause reduction of the symmetry. Is a one degree convergence is negligible.
The authors have to carry out proper simulations sustaining this assumption.
AR: We thank the reviewer for identifying this potential issue. We agree that oblique incidence can cause symmetry reduction, and therefore we have performed full-wave electromagnetic simulations to determine the effect of oblique incidence. We have added a section to the supplementary information to discuss these simulation and compare predicted mode splittings with experimentally measured mode splittings. From this analysis, we find that nonzero angle of incidence is not the largest contributor to symmetry breaking when the degeneracy is purposely lifted through design. The new section in the supplementary information has been reproduced below for the reviewer's convenience.

Effect of nonzero angle of incidence
In the experimental measurements, the angle of incidence was maintained less than 1°. Nonzero angle of incidence results in symmetry breaking. To determine the effect of angle of incidence on symmetry reduction, we performed full-wave electromagnetic simulations of reflectance (Fig. S9a) when the resonator symmetry breaking (S) was 0.4 and lattice deformation (δ) of 0 nm for angles ranging from 0°to 3°. From the calculated reflectance spectra, as the angle of incidence is increased the induced symmetry breaking results in the degeneracy being lifted and a reduction of the Q-factors. The calculated mode splitting (Fig. S9b) was recovered from the simulated reflectance curves by fitting the spectra with two Fano resonances. When the angle of incidence reaches 1°the mode splitting increases to 0.3 nm. Increasing the angle of incidence further to 3°results in a splitting of 2.7 nm. These calculated mode splittings are below the experimentally measured mode splittings for the half-circle, half-square resonator deformation case which observed splittings of 4 nm and 10 nm demonstrating that nonzero angle of incidence is not the largest contributor to symmetry breaking. For additional confirmation, we fit the experimentally measured reflectance measurements (Fig. S9c) of degenerate quasi-BICs when the resonator symmetry breaking (S) was 0.4 and lattice deformation (δ) of 0 nm. Using a two Fano resonance fit resulted in a mode splitting of 1.5 nm representing an upper bound on the effect of angle of incidence since symmetry breaking can be caused by both nonzero angle of incidence and fabrication imperfections. This observed mode splitting would correspond to an angle of incidence of 2.2°for the case when nonzero angle of incidence was the only source of symmetry breaking. Mode Splitting (nm) Figure R4: Effect of nonzero angle of incidence (a) Calculated reflectance spectra when the angle of incidence is nonzero. (b) To calculate the effect of angle of incidence on mode splitting, the calculated reflectance spectra were fit with two Fano resonances. As the angle of incidence was increased from 0°the degeneracy was lifted. (c) Representative experimentally measured reflectance spectra for the case when the modes are degenerate quasi-BICs (S=0.4, δ =0 nm). Fitting the reflectance spectra with two Fano resonances results in a mode splitting of 1.5 nm providing a metric for the amount of symmetry breaking from nonzero angle of incidence and fabrication imperfections.

Comment #4
RC: The manuscript contains vast of the group-symmetry terms, which are in excessive way and sometimes misleading. I recommend the authors to give their manuscript to an expert in the group theory analysis applied to condensed matters to improve the terminology.
AR: The group theory notation we use in our manuscript is chosen to be identical to that of "Group theory and its applications in physics," by Inui, Tanabe, and Onodera, Ref.
[55] in the manuscript, a standard textbook for group theory across all of physics, and in particular in condensed matter physics. Since it is necessary to keep track of space group symmetry in addition to point group symmetry to determine the symmetry of the perturbed field, we have adopted the abbreviated Hermann-Mauguin notation used by the International Union of Crystallography (IUCr), which is consistent with other published works in this field. However, we agree that this standard notation can be slightly frustrating, and so we have endeavored to clarify the language surrounding this notation to improve the manuscript's readability.
This has resulted in numerous changes throughout the manuscript's abstract and main text.

Comment #5
RC: Several times the authors claim on benefits over the problem of fabrication imperfections, however they refer to no studies. For example, recent paper in Nanophotonics 10, 4313 (2021) examines the robustness of the BIC of different type against the typical fabrication imperfectness, that is positional disorder. I recommend the authors to consider that paper and references therein.
AR: We thank the reviewer for pointing out this oversight. We have added appropriate references when discussing the susceptibility of BICs to disorder.
AR: To address the reviewer's comments we have made major revisions to the manuscript. Broadly speaking, these changes included adding discussions and results to demonstrate C 2 -breaking and C 3 -breaking controlling the Q-factors and mode splitting, respectively, which clarifies the design paradigm. Furthermore, we have reorganized the theoretical and experimental results for increased clarity.

Reviewer #3
RC: In the manuscript, Authors discuss the origin of bound states in the continuum, high-Q resonances of periodic photonic structures, in terms of group theory. In particular, they present an approach to create two degenerate symmetry-protected bound states in the continuum and control their quality factor. It is a beautiful example of rigorous theoretical analysis which results can be implemented in practical applications.
AR: We want to thank the reviewer for their time and constructive feedback. The feedback enabled us to significantly strengthen the manuscript.

Comment #1
RC: Line 145 seems to have a misprint.
AR: Thank you for pointing out this awkward wording. We have modified the section in the course of addressing Reviewer #3, Comment #2. Please see Comment #2 for the revised text.
2 modes of a silicon metasurface. When the resonator is deformed into a triangle, only C 2 symmetry is broken resulting in two degenerate quasi-BICs. The Q-factors can be controlled by the resonator symmetry breaking parameter (S). Correspondingly, when the lattice is contracted in one direction two non-degenerate BICs are formed since only C 3 symmetry is broken. With these two symmetry breaking parameters it is possible to design a lattice with cm C s symmetry capable of having non-degenerate quasi-BICs with arbitrary splitting and Q-factors. (b) Field profile for the four symmetry phases.

Comment #3
RC: Could the Authors comment on how the orientation of the axis along which the lattice is squeezed, is chosen? Is it restricted by basis vectors of the primitive unit cell? Will there be any quantitative differences in Qs and frequencies splitting compared to the discussed case?
AR: We thank the reviewer for bringing up this question. While the lattice can be squeezed along any axis that reduces the lattice to a centered rectangular Bravais lattice, the resulting and Q-factors and mode splittings can differ significantly. To address this issue, we have added a section the supplementary information discussing the effect of lattice contraction direction on mode splitting and Q-factors. The section is reproduced below for the reviewer's convenience.

Effect of lattice contraction direction
To understand the role of lattice contraction direction on mode splitting and Q-factors we performed full-wave eigenvalue simulations for three lattice contraction directions: vertical, 63.4°diagonal, and horizontal. A schematic of the three contraction directions with respect to the triangular lattice is presented in Fig. 10a. For all cases, the C 2 symmetry breaking parameter (S) was 0.6. The calculated Q-factors and mode splittings (Fig. 10b) show that the choice of lattice contraction direction can have three quantitative effects. First, for a given mode splitting the mean Q-factors can depend strongly on contraction direction. For example, when the splitting was near 25 nm, the mean Q-factors (Q) for the vertical and horizontal cases are 2700 and 2730 respectively. In comparison, for the diagonal contraction case the calculated Q-factors were significantly higher with a Q of 3380. The second effect is the difference between the maximum (Q max ) and minimum (Q min ) Q-factors (∆Q = |Q max − Q min |).
For the diagonal contraction case, the calculated ∆Q when the mode splitting is close to 25 nm is 1110. This difference reduces to 980 for the horizontal contraction case. The smallest ∆Q is obtained for the vertical contraction with a difference of 920. The final quantitative effect of lattice contraction direction is the degree of symmetry breaking for a given amount of lattice contraction. For the diagonal contraction case, the unit cell area has been reduced the most, but only achieves a mode splitting of around 25 nm. While the unit cells are larger for the vertical and horizontal cases, they achieve mode splittings of 45 nm and 41 nm respectively. The reduced splitting in the diagonal case can be attributed to the fact that the diagonal lattice contraction is closer to preserving the triangular lattice (albeit with a scaled periodicity). If the lattice contraction direction only scaled the periodicity of the triangular lattice, C 3 symmetry would be preserved and the modes would remain degenerate. Combined, these three quantitative effects from lattice contraction direction demonstrate how the choice of symmetry breaking method can significantly affect device performance and provide insight into potential routes to optimize symmetry breaking operations.   Figure R6: Effect of lattice contraction direction on Q-factor (a) Schematic denoting three different directions for contracting the triangular lattice. (b) Calculated Q-factors from full-wave eigenvalue simulations for vertical, horizontal, and diagonal lattice contractions. For equivalent mode splitting values the mean Q-factor (Q) and difference between Q max and Q min can differ significantly.